MURI ADCN Workshop

John Doyle, Steven LowEAS, CaltechOSU, Columbus

October 14, 2010

Post-docsLijun ChenKrister JacobssonNader MoteeChee-Wei Tan

Grad studentsMasoud FarivaJavad LavaeiJK NairSomayeh Sojoudi

Outline Overview of Caltech projects (40 mins,

Low)

Optimal wireless protocols and devices (40 mins, Lavaei)

Overview Heavy-tailed traffic File fragmentation to mitigate heavy-tailed

delay Tail-robust scheduling algorithms

Wireless Random access game Smart antenna design Power control

Congestion control Effect of ack-clocking Reverse-engineering transients

Overview Heavy-tailed traffic File fragmentation to mitigate heavy-tailed

delay Tail-robust scheduling algorithms

Wireless Random access game Smart antenna design (Javad Lavaei) Power control

Congestion control Effect of ack-clocking Reverse-engineering transients

File fragmentation

File fragmentation over an unreliable channelJ. Nair, M. Andreasson, L. Andrew, S. Low and J. Doyle. IEEE Infocom, San Diego, CA, March 2010

File fragmentation: summary Motivation: how to mitigate heavy tail? Recent work showed file transfer time can be

heavy-tailed even if file size is light-tailed(Fiorini, Sheahan, & Lipsky, 2005; Jelenkovic & Tan 2007; etc.)

Results Independent or bounded fragmentation

preserves light-tailedness Constant fragmentation min expected delay Asymptotically optimal design: blind

fragmentation Optimal or blind fragmentation preserves tail

index

Model Given file of random size L L is fragmented into K packets for

transmission at unit rate n-th transmission of size

n-th transmission is successful if

where are iid with distribution F

nxfile fragment constant overhead

nn xA

nA

Model

LlxAxll nnnnn

1

1 )( 1

remaining file size at time n+1

fragment size at n

per-packet overhead

iid random var of distr F

Model

LlxAxll nnnnn

1

1 )( 1

per-stage cost: )0()( nnn lx 1

total cost:

11

)0( )()(n

nnn

n lxLT 1

Prior work

1

1

1

)0( )()(

)(

nnn

nnnnn

lxLT

LlxAxll

1

1

Theorem [Fiorini, Sheahan, & Lipsky, 2005; Jelenkovic & Tan 2007]

Without fragmentation T(L) has heavy tail even when L is light-tailed,

provided F has unbounded support

nLxn

Result: LT-preserving frag

independent fragmentation: nnnn XlXx iid ,,min

boundedfragmentation:

TheoremWith independent frag or bounded frag:T(L) is light-tailed provided L is light-tailed

Then, heavy-tailed delay originates only from heavy-tailed files

nn lbxb ,min

Result: optimal fragmentation

TheoremConstant fragmentation is uniquely optimal• Optimal #fragments: K*(L) =

• Optimal fragment size: x*(L) = L/K*(L)

•

per-bit cost:)(

)(

xFx

xxg

)(minarg0

xgax

aLinteger

LaaLx

Laa

/1)(

/1*

)( min LTxE

Result: blind fragmentation

Theorem• for all L• Blind fragmentation is asymptotically

optimal

)(minarg0

xgax

LaLx as )(*

blind fragmentation: nn lax ,min

)()()( * aagLJLJ a

expected total cost: )(:)( LTLJ xx E

Result: tail distribution of T(L)

Theorem• If L light-tailed, so is T(L) • If L RV(a) (heavy-tailed), so is T(L)

)(~)(

)(~)(*

agtLPtLTP

agtLPtLTP

a

optimal frag: )(,...,1 ),( ** LKnLxxn

blind frag: nn lax ,min

Overview Heavy-tailed traffic File fragmentation to mitigate heavy-tailed

delay Tail-robust scheduling algorithms

Wireless Random access game Smart antenna design (Javad Lavaei) Power control

Congestion control Effect of ack-clocking Reverse-engineering transients

Tail-robust scheduling

Tail-robust scheduling via Limited Processor SharingJ. Nair, A. Wierman, and B. Zwart.Proc. IFIP Performance, 2010; to appear in Performance Evaluation

The “simplest” scheduling model

Q: What policy minimizes mean response time?A: Shortest Remaining Processing Time (SRPT)

RobustOptimal regardless of interarrival times, job sizes, etc.

A Wierman

Q: Can a policy be optimal & robust for the tail?

Power-law job sizes Light-tailed job sizesWe’ll study the decay rate:

log P( )( ) limt

X tγ Xt

We’ll study the tail index:log ( )Γ( ) lim

logt

P X tXt

Lot’s of analysis over the last 20+ years…

Γ( )P( ) XX t t ( )P( ) γ X tX t e A Wierman

SRPT Optimal [NWZ 08] Worst possible [NZ 06]Optimal [BBQZ 06] Worst possible [MZ 06]Worst possible [B76] Optimal [RS 01]Optimal [MT 80] Worst possible [NWZ 08]Worst possible [A99] Worst possible [N 07]

Power-law sizes Light-tailed sizes

Q: Can a policy be optimal & robust for the tail?

PSFCFSPLCFSLCFS

Lot’s of analysis over the last 20+ years…

A Wierman

^(non-learning)

A: NO!Q: Can a policy be optimal & robust for the tail?

A Wierman

Theorem: There does not exist a work-conserving,online, non-learning scheduling policy ν that has:

for all ε>0 and work-conserving, online policies πunder both light-tailed and power-law job sizes.

1( )limsup( )

εν

x π

P T xP T x

Corollary:Optimal under power-laws worst-case under light-tails,and vice-versa

A Wierman

Q: Can a policy be optimal & robust for the tail? ^

(non-learning)

A: NO!

Q: Can a policy be weakly robust for the tail? ^

(non-learning)

better-than-worst-case under bothlight-tailed and power-law workloads

A: No known policies are.

A Wierman

Our candidate: Limited Processor Sharing, LPS(c)

PSFCFS queue at most c jobs

is weakly robust and optimal for large classes of power-law and light-tailed distributions.

1 11

cρ

…but it uses ρ

A Wierman

c=1FCFS

c=∞PS

Power-law

Light-tailed

Response time tail gets lighter

Response time tail gets lighter

c

c

A Wierman

c=1FCFS

c=∞PS

Light-tailed

kc ≥ 2

better-than-worst-case

better-than-worst-case

Power-law

c < ∞

A Wierman

c=1FCFS

c=∞PS

Light-tailed

better-than-worst-case

better-than-worst-case

optimal (if sizes have a finite variance)

optimal (if sizes are more “variable” thanan Exponential dist.)

1 11

cρ

Power-law

A Wierman

Overview Heavy-tailed traffic File fragmentation to mitigate heavy-tailed

delay Tail-robust scheduling algorithms

Wireless Random access game Smart antenna design (Javad Lavaei) Power control

Congestion control Effect of ack-clocking Reverse-engineering transients

Random access game

Random Access Game and MAC Design, L. Chen, S. H. Low and J. C. Doyle, IEEE/ACM Transactions on Networking, 2010

Contention-based MAC (contention control)

Two components A contention resolution algorithm: adjusts channel

access probability in response to the contention A feedback mechanism: updates a contention

measure and sends it back to wireless nodes

)(tpi

)(tqi

L. Chen

Dynamical model

The exact form of and are determined by or can be designed for the specific MAC protocol

Present a game-theoretic model to understand the dynamical system (1) and use it to design new protocols

)(tpi

)(tqi

))(()1())(),(()1(

tptqtqtptp

ii

iiii

GF

iF iG(1)

L. Chen

Random access game

),( iiii qpp F

)( iii pFq

iiiii dppFpU )()(

fixed point

Only determined by the contention resolution algorithm Usually continuous, increasing and concave

))(()1())(),(()1(

tptqtqtptp

ii

iiii

GF

Definition: A random access game is defined as a quadruple is a set of players (wireless nodes)

Strategy with

Payoff function with given contention measure

MAC (i.e., system (1)) as strategy update algorithm achieving the equilibrium of random access game The equilibrium properties can be understood and

designed through the specification of and The adaptation of channel access probability can

be specified through , corresponding to different strategies to approach the equilibrium.

})(,)( ,)( ,{ : NiiNiiNii quSN GN

]},[|{ : iiiii wppS 10 ii wv

)()( : )( pqppUpu iiiii (p)q ii G

G

iU iq

G)(F,

Conditional collision probability as contention measure

Assumptions (single cell wireless LANs): A0: is continuously differentiable, strictly

concave, and with bounded curvature away from zero, i.e.,

A1: let and denote the smallest eigenvalue of by . Then, .

A2: functions are all strictly increasing or all strictly decreasing

)1(1 jIji p(p)qi

)(iU

0/1)(/1/1 '' ii pU)1()( ii

pp )(2 p min 0min

))(1)(1()( 'iiiii pUpp

Equilibrium Theorem: Under assumption A0, there exists a

Nash equilibrium for random access game. Suppose additionally A1 holds. Then random access game has a unique Nash equilibrium. A channel access probability is a Nash

equilibrium of random access game, if

Proof: By showing the equilibrium condition

is the optimality condition for a strictly convex optimization problem.

*p

. , ,),(),( *1

** NiSpppuppu iiiiiii

iiiiiii SppppqpU ,0)))(()(( ***'

Nontrivial Nash Equilibrium A Nash equilibrium is a nontrivial equilibrium if

for all nodes , the equilibrium strategy satisfies

and trivial equilibrium otherwise. Theorem: Suppose A2 holds. If the random

access game has a nontrivial Nash equilibrium, it must be unique. Proof by contradiction: Note that a nontrivial Nash

equilibrium

*ipi

),()( **' pqpU iii

., ),()()( *** Njippp jjii

Definition: A Nash equilibrium is said to be symmetric if for all , and an asymmetric equilibrium otherwise. By symmetry, there must have multiple

asymmetric equilibria if there exists any. Theorem: For a system with several classes of

users, suppose A1 and A2 hold. If random access game has a nontrivial equilibrium, it must be unique and symmetric. Guarantees fair sharing of wireless channel among the same

class of wireless nodes Provides service differentiation among different

classes of wireless nodes

*p**ji pp Nji ,

Gradient play

Have a nice economic interpretation Theorem: Suppose A0 and A1 hold. The

gradient play converges to the unique Nash equilibrium of the random access game if for any , the stepsize

Proof by Lyapunov method. Also studied its robust verification to the

estimation error.

isiiiiii tpqtpUttptp )))](())(()(()([)1( '

.1||

2

N

fi

i

A concrete MAC design Consider a single-cell network with classes

of users Each class associated with a weight

Assume Want to achieve maximal throughput under

the weighted fairness constraint

L

l

.,1 , LmlTT

m

l

m

l

l.2max1 L

Utility design Let . Under the assumption of Poisson

arrival, the throughput achieves maximum at that satisfies

the duration of idle slot, the duration of a collision

Under the decoupling approximation, to achieve weighted fairness requires

ii

p

*

ce T/1)1(**

cT

.,1 , Lmlpp

m

l

m

l

Requires

A convenient choice

Utility function

*

)(

)())(1)(1()(

*

'

eph

phpUpp

l

l

l

llllll

)1()(*

l

l

l

l peph

],0[

)1ln()11()1()(*

*

wp

pepepU

l

ll

ll

ll

Equilibrium and dynamics Theorem: Suppose

The random access game has a unique and nontrivial equilibrium

The gradient play converges if the stepsize

./11

1/1

1

maxmax

*

*

*

ewe

e

.1||

2

N

fi

Performance: throughput

Performance: collision

Performance: short-term fairness

Performance: dynamic scenario

Performance: service differentiation

A natural progression

Centralized optimization

Distributed but cooperative actions with rich information and signaling allowed

Less cooperation(economic perspective)

Less information or signaling available(engineering perspective)

Optimization

Game theory

Overview Heavy-tailed traffic File fragmentation to mitigate heavy-tailed

delay Tail-robust scheduling algorithms

Wireless Random access game Smart antenna design (Javad Lavaei) Power control

Congestion control Effect of ack-clocking Reverse-engineering transients